Free Function Theory Through Matrix Invariants
Free Function Theory Through Matrix Invariants
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通过矩阵不变量的自由函数理论
DOI：
10.4153/cjm-2015-055-7
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
S. Spenko
中科院分区：
文献类型：
--
作者：
I. Klep;S. Spenko
摘要 本文涉及自由函数理论。自由映射是多个复变量中解析函数的自由模拟，并根据自由非交换变量进行定义。 $g$ 非交换变量的函数是 $g$ 上所有大小的方阵元组的函数，它尊重直和和联立共轭。此类映射的示例包括非交换多项式、非交换有理函数和收敛非交换幂级数。与自由分析中的现有文献形成鲜明对比，本文研究了具有对合的自由映射，即实解析函数的自由模拟。为了掌握这些，开发了不变理论的技术和工具并将其应用于自由分析。以下是所获得结果的示例。提出了通过其有限维切片的属性来表征多项式自由映射，然后用于建立关于标量和非标量点的解析自由映射的幂级数展开；后者是一系列广义多项式，给出了不变理论表征。此外，还得到了包含对合的自由映射的反函数隐函数定理。最后，通过精心挑选的示例表明，具有对合的自由映射并不表现出无对合映射所具有的强刚性特性。
Abstract This paper concerns free function theory. Free maps are free analogs of analytic functions in several complex variables and are defined in terms of freely noncommuting variables. A function of $g$ noncommuting variables is a function on $g$ -tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions, and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps with involution, free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invariant-theoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involution-free counterparts.
多个非交换变量的全局全纯函数 II
DOI：
10.4153/cmb-2017-044-4
发表时间：
2018
期刊：
Canadian Mathematical Bulletin
影响因子：
--
作者：
Agler, Jim;McCarthy, John
通讯作者：
McCarthy, John
隐函数定理和自由代数集
影响因子：
1.3
作者：
Agler, Jim
通讯作者：
Agler, Jim